# Does the electrodynamics-like PDE $\epsilon^{ijk}\partial_j B_{kl}(x) = \delta^i_l\delta^{(3)}(x)$ have solutions?

Consider the following PDE in 3 dimensions $$\epsilon^{ijk}\partial_j B_{kl}(x) = \delta^i_l\delta^{(3)}(x)$$ Does $$B_{kl}(x)$$ have a solution? (It can have any kind of singularity, e.g. it can have singularity at the positive $$z$$ axis like a Dirac string emanating from the origin.)

• Is $B_{kl}$ antisymmetric? – Qmechanic Nov 15 '18 at 9:01
• There is no such constraints – Weicheng Ye Nov 15 '18 at 12:09
• Then the eq. system decouples in 3 independent subsystems: $l=1$, $l=2$, & $l=3$. – Qmechanic Nov 16 '18 at 12:26